This invention relates to curve generation for computer graphics.
Many computer applications generate or model curves. Computer aided design (CAD) systems, computer animation tools, and computer graphics applications all attempt to replicate smoothly curved real-world objects, or to generate new curved objects designed and developed with these tools. Many objects are not susceptible to exact mathematical description, and are often modeled interactively by a user employing artistic instead of scientific criteria. Computer systems require satisfactory methods of representing these objects and their edges and surfaces. Since computers have finite storage and processing capacity, an object cannot be modeled with an infinite number of coordinate points. Instead, object surfaces are approximated by segments such as curves, lines, planes, and other object "primitives" that are easier to describe mathematically.
Certain modeling methods use parametric polynomial curves to build up a model of an object. Typically, polynomial curve methods employ parametric equations based on cubic polynomial equations of a parameter (for a line segment, one parameter is used, e.g., "s"; for a surface, two parameters are used, e.g., "u" and "v"). A number of different polynomial forms for lines and surfaces have been developed, including Hermite, Bezier, and B-spline. Whatever parameter form is chosen, one or more curved lines or surfaces can be generated based upon a compact set of control points (or vertices) that unambiguously define the shape of those curves or surfaces. Parametric curves can be "re-parameterized" by a substitution of one ordered set of parameters for another in a one-for-one transformation, without changing the curve's appearance.
Often, a computer model of an object will require that several separately created curves match at their boundaries to create one seamless integrated object.